The flat map of a two-form on a manifold #
A two-form σ on a manifold M gives at each point its flat map
σ♭_x : T_x M → T_x^* M, v ↦ ι_v σ = σ(v, ·) (TauCeti.SmoothTwoForm.flat), the continuous
form of the algebraic bilinear form TauCeti.SmoothForm.bilinFormAt. The flat maps of a smooth
two-form form a smooth section of the bundle Hom(TM, Hom(TM, ℝ))
(TauCeti.SmoothTwoForm.contMDiff_flat). On a finite-dimensional manifold, the form is
nondegenerate exactly when every flat map is invertible
(TauCeti.SmoothTwoForm.isNondegenerate_iff_isInvertible_flat); the inverse maps then form a
smooth section as well, by ContMDiff.clm_bundle_inverse.
Inverting the flat map turns covector fields into vector fields: a Hamiltonian vector field is
(σ♭)⁻¹ df, and the Liouville field of a primitive α is (σ♭)⁻¹ α. For a nondegenerate form,
in any dimension, a vector field is determined by its contraction with the form
(TauCeti.SmoothTwoForm.IsNondegenerate.interior_injective), and a linear map between two tangent
spaces of M that preserves the form is surjective in finite dimensions
(TauCeti.SmoothTwoForm.IsNondegenerate.surjective_of_eval₂_map).
On smooth sections, contraction X ↦ ι_X σ is a linear map from smooth vector fields to smooth
one-forms (TauCeti.SmoothTwoForm.interiorₗ). For a nondegenerate form on a finite-dimensional
manifold it is a linear equivalence (TauCeti.SmoothTwoForm.IsNondegenerate.interiorEquiv), whose
inverse sends a one-form α to the vector field (σ♭)⁻¹ α.
Main declarations #
TauCeti.SmoothTwoForm.flat: the flat mapv ↦ σ(v, ·)at a point.TauCeti.SmoothTwoForm.interior_apply_eq_ofSubsingleton_flat: the interior productι_X σis the flat map applied toX.TauCeti.SmoothTwoForm.IsNondegenerate.interior_injective: for a nondegenerate form,X ↦ ι_X σis injective.TauCeti.SmoothTwoForm.contMDiff_flat: the flat maps form a smooth section.TauCeti.SmoothTwoForm.contMDiffAt_sectionAlong_flatL: the flat maps of a smooth family of two-forms along a map depend smoothly on the parameter.TauCeti.SmoothTwoForm.isNondegenerate_iff_isInvertible_flat: nondegeneracy is invertibility of the flat maps, in finite dimensions.TauCeti.SmoothTwoForm.IsNondegenerate.surjective_of_eval₂_map: a linear map between tangent spaces preserving a nondegenerate form is surjective, in finite dimensions.TauCeti.SmoothTwoForm.interiorₗ: contraction of smooth vector fields, as a linear map to smooth one-forms.TauCeti.SmoothTwoForm.IsNondegenerate.interiorEquiv: for a nondegenerate form in finite dimensions, contraction identifies smooth vector fields with smooth one-forms.
References #
- D. McDuff, D. Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford, 2017, Section 3.1.
The flat map of a two-form at a point: the contraction v ↦ ι_v σ = σ(v, ·), as a
continuous linear map from the tangent space to its dual.
Instances For
The flat map evaluates as the two-form.
The interior product ι_X σ at a point is the flat map applied to X, read as a one-form.
For a nondegenerate two-form, a vector field is determined by its contraction with the form:
if ι_X σ = ι_Y σ, then X = Y. In particular, the Liouville field of a primitive is unique.
Flat maps of a smooth family of two-forms. Let σ m be alternating two-forms on the
tangent spaces T_(b m) M along a map b : N → M, depending C^n on m. Then their flat maps
v ↦ σ m (v, ·) depend C^n on m, as a section of Hom(TM, Hom(TM, ℝ)) along b.
The flat maps of a two-form form a smooth section of the bundle Hom(TM, Hom(TM, ℝ)).
The flat map is the continuous form of the algebraic bilinear form of σ at a point.
In finite dimensions, a two-form is nondegenerate exactly when all its flat maps are invertible.
The flat maps of a nondegenerate two-form on a finite-dimensional manifold are invertible.
Linear maps preserving a nondegenerate form are surjective. A linear map D from the
tangent space at x to the tangent space at y with σ_y(D u, D v) = σ_x(u, v) is injective
because σ is nondegenerate, hence surjective in finite dimension.
Contraction of smooth vector fields #
Contraction X ↦ ι_X σ = σ(X, ·) of smooth vector fields into a two-form, as a linear map to
smooth one-forms.
Instances For
The one-form ι_X σ is the interior product of σ with X.
As a section of the bundle of covectors, ι_X σ is the flat map applied to X.
For a nondegenerate two-form on a finite-dimensional manifold, contraction X ↦ ι_X σ is a
linear equivalence between smooth vector fields and smooth one-forms. Its inverse sends a one-form
α to the vector field (σ♭)⁻¹ α.
Instances For
The nondegenerate contraction equivalence is contraction.
The vector field contracting to a one-form α is (σ♭)⁻¹ α.
The vector field (σ♭)⁻¹ α contracts to α.